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Israel has a border on the east, northeast and the Gaza Strip borders the northeast corner of the Sinai peninsula.
In modern Israel, it's the city of Eilat and the nearby Taba border-point. The boundary is the border with Egypt's Sinai area.
Almost every war between Egypt and Israel has been fought in or using the Sinai Peninsula. Egypt has used the Sinai Peninsula as a launching point for its invasion of Israel in 1948 and in preparations for the likely invasion of Israel in 1967. Israel conquered the Sinai and was forced to relinquish it in 1956. In 1973, the Sinai Peninsula (under Israeli Occupation) served as a strong buffer to the Egyptian surprise invasion attempt.
Anwar Sadat (أنور السادات), the Prime Minister of Egypt, concluded a peace treaty with Israel, got Israel to return the Sinai Peninsula and its oil resources to Egyptian sovereignty in the process, and was assassinated by Egyptian militants for his efforts.
None now. Israel used to own the Sinai Peninsula which is part of Africa, but now the Sinai is back in Egyptian control.
Israel agreed to return the Sinai Peninsula to Egyptian control in three phases (from 1979 to 1982) in return for recognition of Israel by Egypt.
Mount Sinai is not in Israel. It is in Egypt, on the southern part of the Sinai Peninsula.
The Sinai Desert borders Israel.
All of those places border Israel.
Israel was invaded. However, it is worth noting that during the Arab-Israeli War of 1948-49, some Israeli divisions crossed into sovereign Egyptian territory (east Sinai) after taking back the Negev Desert in the south of Mandatory Palestine from Egyptian forces. When Israel retreated from the Sinai, Egypt was willing to sign an armistice.
Israel captured the Sinai Peninsula and the Gaza strip during the 1967 war. The Sinai Peninsula was reverted to Egypt under the 1979 peace treaty. Israel evacuated its settlers and military from the Gaza strip in 2005, and handed it over to the PLO. The residents of the strip trashed the economic infrastructure inherited from Israeli settlements, and Hamas expelled the PLO. There is presently (early 2011) no 'former' Egyptian territory occupied by Israel.
If you cross the border into Sinai in Eilat/Taba, 85 shekel plus 56 Egyption Lira (assuming you have no car), will get you a 14 days visa to western sinai only (up to Sharem-a-Shiekh). Please note that as of Sept. 2008, there are severe warnings regarding El-Quaida and Hamas terrorists aiming at western tourists in Sinai. Take care.